Journal of Function Spaces
Volume 2015 (2015), Article ID 980352, 22 pages
http://dx.doi.org/10.1155/2015/980352

Research Article

Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

L. C. Ceng,¹ A. Latif,² C. F. Wen,³ and A. E. Al-Mazrooei⁴

¹Department of Mathematics, Shanghai Normal University, and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
²Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
³Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 807, Taiwan
⁴Department of Mathematics, College of Science, University of Jeddah, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 28 October 2014; Accepted 5 January 2015

Academic Editor: Mohamed-Aziz Taoudi

Copyright © 2015 L. C. Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

1. Introduction

Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called -Lipschitz continuous if there exists a constant such that

In particular, if , then is called a nonexpansive mapping; if , then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP): find a point such that

The solution set of VIP 2 is denoted by .

The VIP 2 was first discussed by Lions [1]. In 1976, Korpelevich [2] proposed an iterative algorithm for solving the VIP 2 in Euclidean space :

with a given number, which is known as the extragradient method.

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [3] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that

We denote the set of solutions of GMEP 4 by .

It was assumed in [3] that is a bifunction satisfying conditions (A1)–(A4) and is a lower semicontinuous and convex function with restriction (B1) or (B2), where

(A1) for all ;(A2) is monotone, that is, for any ;(A3) is upper-hemicontinuous; that is, for each ,

(A4) is convex and lower semicontinuous for each ;(B1) for each and , there exists a bounded subset and such that, for any ,

(B2) is a bounded set.

For a given positive number , let be the solution set of the auxiliary mixed equilibrium problem; that is, for each ,

On the other hand, let be a single-valued mapping of into and let be a multivalued mapping with . Consider the following variational inclusion: find a point such that

We denote by the solution set of the variational inclusion 8. It is known that problem 8 provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory. Let be a maximal monotone set-valued mapping. We define the resolvent operator associated with and as follows:

where is a positive number.

Let be two mappings. Consider the following general system of variational inequalities (GSVI) of finding such that

where and are two constants. It was considered and studied in [4–6]. In particular, if , then the GSVI 10 reduces to the following problem of finding such that

which is defined by Verma [7] and it is called a new system of variational inequalities (NSVI). Further, if , additionally, then the NSVI reduces to the classical VIP 2. In 2008, Ceng et al. [6] transformed the GSVI 10 into a fixed point problem in the following way.

Proposition CWY (see [6]). For given , is a solution of the GSVI 10 if and only if is a fixed point of the mapping defined by

where .

In particular, if the mapping is -inverse strongly monotone for , then the mapping is nonexpansive provided for . We denote by the fixed point set of the mapping .

Let and be two nonexpansive mappings. In 2009, Yao et al. [8] considered the following hierarchical VIP: find hierarchically a fixed point of , which is a solution to the VIP for monotone mapping ; namely, find , such that

The solution set of the hierarchical VIP 13 is denoted by . It is not hard to check that solving the hierarchical VIP 13 is equivalent to the fixed point problem of the composite mapping ; that is, find , such that . The authors [8] introduced and analyzed the following iterative algorithm for solving the hierarchical VIP 13:

Theorem YLM (see [8, Theorem ]). Let be a nonempty closed convex subset of a real Hilbert space . Let and be two nonexpansive mappings of into itself. Let be a fixed contraction with . Let and be two sequences in . For any given , let be the sequence generated by 14. Assume that the sequence is bounded and that

(i);(ii), ;(iii), and ;(iv);(v)there exists a constant such that for each , where . Then, converges strongly to which solves the hierarchical VIP

Very recently, Iiduka [9, 10] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with bilevel programming problems or hierarchical constrained optimization problems or hierarchical fixed point problem, it is refereed as triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such a problem. The convergence analysis of the proposed algorithms was also studied in [9, 10]. Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical variational inequality (THVI). Subsequently, Zeng et al. [11] introduced and considered the following triple hierarchical variational inequality (THVI).

Problem I. Assume that

(i)each is a nonexpansive mapping with ;(ii) is -inverse strongly monotone;(iii) is -strongly monotone and -Lipschitz continuous;(iv).

Then, the objective is to

In [11], the authors proposed the following algorithm for solving Problem I.

Algorithm ZWY ([11, Algorithm  3.2]). Let and satisfy assumptions (i)–(iv) in Problem I. The following steps are presented for solving Problem I.

Step  0. Take , , and , choose arbitrarily, and let .

Step  1. Given , compute as

where , for integer , with the mod function taking values in the set ; that is, if for some integers and , then if and if .

The following convergence analysis was presented in [11].

Theorem ZWY ([11, Theorem ]). Let , , and , such that (i) , (ii) , (iii) or , (iv) or , and (v) for all . Assume in addition that

Then, the sequence generated by Algorithm ZWY satisfies the following properties:

(a) is bounded;(b) and ;(c) converges strongly to the unique solution of Problem I provided .

In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of finitely many GMEPs, finitely many variational inclusions, and general system of variational inequalities.

Problem II. Let be two integers. Assume that

(i) is a sequence of nonexpansive self-mappings on and is -inverse strongly monotone for ;(ii) is -inverse strongly monotone and is -strongly monotone and -Lipschitz continuous;(iii) is a bifunction from to satisfying (A1)–(A4) and is a proper lower semicontinuous and convex function, where ;(iv) is a maximal monotone mapping, and and are -inverse strongly monotone and -inverse strongly monotone, respectively, where and ;(v), where .

Then, the objective is to

Motivated and inspired by the above facts, we introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove that under mild conditions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings , the solution set of finitely many GMEPs, the solution set of finitely many variational inclusions, and the solution set of GSVI 10, which is just a unique solution of the THVI 19. In addition, we also consider the application of the proposed algorithm to solve a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and GSVI 10. That is, under appropriate conditions, we prove that the proposed algorithm converges strongly to a unique solution of the VIP: , ; equivalently, . The results obtained in this paper improve and extend the corresponding results announced by many others including [12, 13]. A comprehensive survey on triple hierarchical variational inequalities can be found in [14].

2. Preliminaries

Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence ; that is,

Recall that a mapping is called

(i)monotone if

(ii)-strongly monotone if there exists a constant such that

(iii)-inverse strongly monotone if there exists a constant such that

It is obvious that if is -inverse strongly monotone, then is monotone and -Lipschitz continuous. Moreover, we also have that, for all and ,

So, if , then is a nonexpansive mapping from to .

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point , satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 1. For given and :

(i), ;(ii), ;(iii), .

Consequently, is nonexpansive and monotone.

Definition 2. A mapping is said to be
(a) nonexpansive if

(b) firmly nonexpansive if is nonexpansive or, equivalently, if is -inverse strongly monotone (-ism),

alternatively, is firmly nonexpansive if and only if can be expressed as

where is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if is nonexpansive, then is monotone. It is also easy to see that a projection is -ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.

Next, we list some elementary conclusions for the mixed equilibrium problem (MEP) where is the solution set.

Proposition 3 (see [15]). Assume that satisfies (A1)–(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:

for all . Then, the following hold:

(i)for each is nonempty and single-valued;(ii) is firmly nonexpansive; that is, for any ,

(iii);(iv) is closed and convex;(v), for all and .

We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 4. Let be a real inner product space. Then, there holds the following inequality:

Lemma 5. Let be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition 1(i)) implies

Lemma 6 (see [16, demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then, is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here, is the identity operator of .

Let be an infinite family of nonexpansive self-mappings on and let be a sequence of nonnegative numbers in . For any , define a mapping on as follows:

Such a mapping is called the -mapping generated by and .

Lemma 7 (see [17, Lemma 3.2]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and , the limit exists where is defined as in 33.

Lemma 8 (see [17, Lemma ]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a sequence of nonexpansive self-mappings on such that , and let be a sequence in for some . Then, .

Let be a nonempty closed convex subset of a real Hilbert space . We introduce some notations. Let be a number in and let . Associating with a nonexpansive mapping , we define the mapping by

where is an operator such that, for some positive constants , is -Lipschitzian and -strongly monotone on ; that is, satisfies the conditions:

for all .

Lemma 9 (see [18, Lemma 3.1]). is a contraction provided ; that is,

where .

Lemma 10 (see [18]). Let be a sequence of nonnegative numbers satisfying the conditions

where and are sequences of real numbers such that

(i) and or, equivalently,

(ii), or .

Then, .

Lemma 11 (see [16]). Let be a real Hilbert space. Then, the following hold:

(a) for all ;(b) for all and with ;(c)If is a sequence in such that , it follows that

Finally, recall that a set-valued mapping is called monotone if, for all , and imply

A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for for every implies . Let be a monotone, -Lipschitz-continuous mapping and let be the normal cone to at ; that is,

Define

Then, is maximal monotone (see [19]) such that

Let be a maximal monotone mapping. Let be two positive numbers.

Lemma 12 (see [20]). There holds the resolvent identity

Remark 13. For , there holds the following relation:

In terms of Huang [21], there holds the following property for the resolvent operator .

Lemma 14. is single-valued and firmly nonexpansive; that is,

Consequently, is nonexpansive and monotone.

Lemma 15 (see [22]). Let be a maximal monotone mapping with . Then, for any given , is a solution of problem 12 if and only if satisfies

Lemma 16 (see [23]). Let be a maximal monotone mapping with and let be a strongly monotone, continuous, and single-valued mapping. Then, for each , the equation has a unique solution for .

Lemma 17 (see [22]). Let be a maximal monotone mapping with and let be a monotone, continuous, and single-valued mapping. Then, for each . In this case, is maximal monotone.

3. Main Results

In this section, we will introduce and analyze a relaxed iterative algorithm for finding a solution of the THVI 19 with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and GSVI 10 in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove the strong convergence of the proposed algorithm to a unique solution of THVI 19 under suitable conditions. In addition, we also consider the application of the proposed algorithm to solve a hierarchical VIP with the same constraints.

We are now in a position to state and prove our first main result.

Theorem 18. Let be a nonempty closed convex subset of a real Hilbert space . Let , be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let be a maximal monotone mapping and let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where , . Let be a sequence of nonexpansive self-mappings on and be a sequence in for some . Let be -inverse strongly monotone for . Let be -inverse strongly monotone with and let be -strongly monotone and -Lipschitz continuous. Assume that either (B1) or (B2) holds and that where . Let and and be sequences in . For arbitrarily given , let be a sequence generated by

where is defined as in 12 with for , and is the -mapping defined by 33. Suppose that the following conditions are satisfied:

(H1), and ;(H2) and ;(H3) and ;(H4) and for all ;(H5) and for all .

Then, there hold the following:

(i);(ii);(iii) provided additionally.

Proof. Let . Taking into account that , we may assume, without loss of generality, that and for all . Since is -Lipschitz continuous, we get

Putting for all , we have

Moreover, observe that

(This is Mann’s iteration method.) Since and , we know from the convexity of that . Put

for all and ,

for all , , and , where is the identity mapping on . Then, we have that and .
We divide the rest of the proof into several steps.
Step  1. We prove that is bounded.
Indeed, utilizing 24 and Proposition 3(ii), we have

Utilizing 24 and Lemma 14, we have

Combining 54 and 55, we have

Since is -inverse strongly monotone for , and for , we deduce that, for any ,

Since is -inverse strongly monotone and , we have

Utilizing Lemma 9, the nonexpansivity of and the one of (due to Proposition CWY), we obtain from 24 and 56 that

where . By induction, we find that

Thus, is bounded and so are the sequences , , , and .
Step  2. We prove that .
Indeed, utilizing 24 and 45, we obtain that

where

for some and for some . Hence, it follows from 24 and that

Also, utilizing Proposition 3(ii), (v), we deduce that

where is a constant such that, for each ,

In the meantime, from 33, since , , and are all nonexpansive, we have

where is a constant such that for each . So, utilizing 24, 63–66, from , , , and , we deduce that

and, by Lemma 9,

where for some . Consequently,

where for some . From (H1)–(H5), it follows that and

Thus, applying Lemma 10 to 69, we immediately conclude that

So, from (H3), it follows that

Step  3. We prove that , , and , where .
Indeed, utilizing Lemma 4, from 48, we get

On the other hand, observe that

Combining 73–75, we get

which leads to

Since , , , and are bounded sequences, it follows from and that

for all and .
Furthermore, by Proposition 3(ii) and Lemma 11 (a), we have

which implies that

By Lemma 11 (a) and Lemma 14, we obtain

which implies

Combining 73 and 82, we conclude that

which yields

Since , , and and , , and are bounded sequences, it follows from 78 and that

Also, combining 55, 73, and 80, we deduce that

which leads to

Since , , and and , and, are bounded sequences, it follows from 78 and that

Hence, from 85 and 88, we get

respectively. Thus, from 89 and 90, we obtain

Next, for simplicity, we write , , and for all . Then,

We now show that ; that is, . As a matter of fact, from 56, 57, and 73 it follows that

which immediately yields

Since , , and and , , and are bounded sequences, it follows from , , that

In the meantime, in terms of the firm nonexpansivity of and the -inverse strong monotonicity of for , we obtain from , , and 57 that

Thus, we have

respectively. Consequently, from 56, 73, and 98, it follows that

which yields

Since , , and and , , , , and are bounded sequences, it follows from 95 that

Also, from 56, 57, 73, and 97, it follows that

which leads to

Since , , and and , , , and are bounded sequences, it follows from 95 that

Note that

Hence, from 101 and 104, we get

Also, observe that and

Hence, from 91, , , and , we obtain that

So, from 106 and 108, we deduce that

In addition, it is clear that

Thus, we conclude from [24, Remark 3.2], (3.28,) and the boundedness of that

Step  4. We prove that .
Indeed, since is reflexive and is bounded, there exists at least a weak convergence subsequence of . Hence, it is known that . Now, take an arbitrary . Then, there exists a subsequence of such that . From 85–89, 91, and 108, we have that , , , , and , where and . Utilizing Lemma 6, we deduce from , 106, and 111 that and (due to Lemma 8). Next, we prove that . As a matter of fact, since is -inverse strongly monotone, is a monotone and Lipschitz continuous mapping. It follows from Lemma 17 that is maximal monotone. Let ; that is, . Again, since , , , we have

That is,

In terms of the monotonicity of , we get

and hence

In particular,

Since (due to 85) and (due to the Lipschitz continuity of ), we conclude from and that

It follows from the maximal monotonicity of that ; that is, . Therefore, . Next, we prove that . Since , , , we have

By (A2), we have

Let for all and . This implies that . Then, we have

By 88, we have as . Furthermore, by the monotonicity of , we obtain . Then, by (A4), we obtain

Utilizing (A1), (A4), and 121, we obtain

and hence

Letting , we have, for each ,

This implies that and hence . Thus, . Consequently, . This shows that .
Step  5. We prove that provided additionally.
Indeed, take an arbitrary . Then, there exists a subsequence of such that . Utilizing the arguments similar to those of 56 and 57, we can readily obtain that for all

Since is -inverse strongly monotone, from 48 and 125 we conclude that for all

which implies that

So, from and the assumption , we get

Thus, it follows from 91 that for all

That is,

Since is -inverse strongly monotone, by Minty’s Lemma [16], we know that 130 is equivalent to the VIP

This shows that . Therefore, .

Theorem 19. Assume that there hold all the conditions in Theorem 18. Then, we have

(i) converges strongly to a point , which is a unique solution of the VIP

(ii) converges strongly to a unique solution of THVI 19 provided additionally.

Proof. Since is -strongly monotone and -Lipschitz continuous, there exists a unique solution of the VIP

Now, let us show that

Since is bounded, we may assume, without loss of generality, that there exists a subsequence of such that and

In terms of Theorem 18 (ii), we know that . So, from 133, it follows that

Next, let us show that . In fact, utilizing Lemma 4, from 48 and 125 with , we get

where . Since , , and (due to 136), we deduce that and

Therefore, applying Lemma 10 to 137, we infer that .
Finally, we prove that provided additionally, where .
Indeed, first of all, let us show that . As a matter of fact, take an arbitrary . Then, there exists a subsequence of such that . Moreover, by Theorem 18 (iii), we know that . Utilizing the arguments similar to those of 56 and 57, we can readily obtain that for all

Utilizing Lemma 4, from 48 and 139, we deduce that for all

where . So, it follows that

Since , and , we find that

Hence, we conclude from 141 that for all

That is,

Since is -strongly monotone and -Lipschitz continuous, by Minty’s Lemma [16], we know that 144 is equivalent to the VIP

This shows that . Taking into account , we know that . Thus, ; that is, .
Next we prove that . As a matter of fact, utilizing 139 and 140 with , we get

Since , , and (due to ), we deduce that and

Therefore, applying Lemma 10 to 146, we infer that . This completes the proof.

Remark 20. It is obvious that iterative scheme 48 is very different from Yao et al. iterative one 14 and Zeng et al. iterative one in Algorithm ZWY. Here, the two-step iterative scheme in [11, Algorithm 3.2] is extended to develop four-step iterative scheme 48 for the THVI 19 by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. It is worth pointing out that under the lack of the assumptions similar to those in [8, Theorem 3.2], for example, is bounded, , and , , for some , the sequence generated by 48 converges strongly to a point , which is a unique solution of the VIP: , .

Remark 21. Theorems 18 and 19 improve, extend, supplement, and develop Yao et al. [8, Theorems 3.1 and 3.2] and Zeng et al. [11, Theorem 3.2] in the following aspects.

(a) THVI 19 with the unique solution satisfying

is more general than the problem of finding a point satisfying in [8] and than the problem of finding a point satisfying in [11, Theorem 3.2]. It is worth pointing out that is nonexpansive if and only if the complement is -inverse strongly monotone; see [25].

(b) Four-step iterative scheme 48 for THVI 19 is more flexible, more advantageous, and more subtle than Zeng et al. two-step iterative one in [11, Algorithm 3.2] and than Yao et al. two-step iterative one 14 because it can be used to solve several kinds of problems, for example, the THVI, the hierarchical VIP, and the problem of finding a common point of four sets: , , , and . In addition, it also drops the crucial requirements that and , , for some in [8, Theorem 3.2 (v)].

(c) The argument techniques in Theorems 18 and 19 are very different from the argument ones in [8, Theorems 3.1 and 3.2] and from the argument ones in [11, Theorem 3.2] because we use the -mapping approach to fixed points of infinitely many nonexpansive mappings (see Lemmas 7 and 8), the properties of resolvent operators and maximal monotone mappings, the fixed point equation equivalent to the GSVI 10 (see Proposition CWY), and the contractive coefficient estimates for the contractions associated with nonexpansive mappings (see Lemma 9).

(d) Compared with the restrictions on the parameter sequences of [8, Theorem 19] and [11, Theorem 3.2], respectively, the hypotheses (H3)–(H5) in Theorem 18 are additionally added because Theorem 18 involves the quite complex problem, that is, the THVI 19 (over the set ) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and GSVI 10.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.

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